to be matched with known quadruped/quadriflagellate gaits.
For a basal morphology of a rectangularly symmetric network [1]} with distinct lengthwise and crosswise couplings
– in Fig. REF  (phase coordination) represented by lines of differing lengths,
spatial symmetries include \(\iota \)  (the identity, fix everything), \(\sigma = (12)(34)\)  (reflect in \(y\) -axis),
\(\rho = (13)(24)\)  (reflect in \(x\) -axis), and \(\sigma \rho = (14)(23)\)  (interchange of diagonals).
In Fig. REF , we attach named gaits and associated symmetry groups to the species in which they are observed.
Additionally some quadriflagellates display a `stand' gait (a transient rest phase where no flagellum beats); this
has the largest number of symmetries: \([\iota ; \phi ], [\sigma ; \phi ],[\rho ; \phi ],[\sigma \rho ; \phi ]\) , for arbitrary \(\phi \) .
